ROBOTICS-SECOND-QUARTER-PPT-5-BINARY-ARITHMETIC-AND-BOOLEAN-ALGEBRA

Boolean Arithmetic

Introduction to Boolean Arithmetic

  • Boolean algebra utilizes two binary values: 0 and 1.

  • Basic operations are defined similarly to regular arithmetic, but with fundamental differences.

Basic Addition in Boolean Arithmetic

  • 0 + 0 = 0

  • 0 + 1 = 1

  • 1 + 0 = 1

  • 1 + 1 = 1

OR Gate

Connection to OR Logic

  • The addition results mirror the truth table for an OR gate:

    • 0 + 0 = 0

    • 0 + 1 = 1

    • 1 + 0 = 1

    • 1 + 1 = 1

Implication

  • Boolean addition acts similarly to the operation of an OR gate and parallel switch contacts.

Operations

Subtraction and Division

  • Subtraction: Not defined in Boolean algebra; it relies on negative values.

  • Division: Not applicable; it relates to subtraction just as multiplication does to addition.

AND Gate

Import of Multiplication in Boolean Algebra

  • Valid operations include multiplication:

    • 0 x 0 = 0

    • 0 x 1 = 0

    • 1 x 0 = 0

    • 1 x 1 = 1

Connection to AND Logic

  • The multiplication results resemble the truth table for an AND gate.

Boolean Variables

Representation of Variables

  • Variables are expressed using capital letters (e.g., A, B).

  • Each variable has a complement: If A=0, A' (or A-bar) = 1, and vice versa.

  • Denoted as:

    • If A=0, then A' = 1.

    • If A=1, then A' = 0.

NOT Gate

Complements

  • The complement of A is written as A-not or A-bar, indicating its opposite value.

  • Use of a prime symbol (A') can also denote complement.

Boolean Addition and Multiplication Rules

Fundamental Rules

  • Boolean addition and multiplication have established identities and rules:

    • Boolean addition is equivalent to OR.

    • Boolean multiplication is equivalent to AND.

    • Complements correspond to the NOT function.

Boolean Algebraic Identities

Definition of Identities

  • An identity holds true for all variable values.

  • For example: X + 0 = X

Additive Identities

  1. Adding Zero: A + 0 = A

  2. Adding One: A + 1 = 1

  3. Adding a Quantity to Itself: A + A = A

  4. Adding a Quantity to Its Complement: A + A' = 1

Multiplicative Identities

Establishing Multiplicative Identities

  1. Multiplying by Zero: A x 0 = 0

  2. Multiplying by One: A x 1 = A

  3. Multiplying a Quantity by Itself: A x A = A

  4. Multiplying a Quantity by Its Complement: A x A' = 0

Properties of Boolean Algebra

Fundamental Properties

  1. Commutative Property:

  • A + B = B + A

  • A x B = B x A

  1. Associative Property:

  • A + (B + C) = (A + B) + C

  • A x (B x C) = (A x B) x C

  1. Distributive Property:

  • A(B + C) = AB + AC

Simplification Rules

Simplification in Boolean Algebra

  • Boolean algebra is frequently used for simplifying logic circuits by translating symbolic form to logic functions and reducing complexity.

Simplification Examples

  1. A + AB = A

  2. A + AB = A + B

  3. (A + B)(A + C) = A + BC

Final Tips

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